Quantum algorithms for zero-sum games

Abstract

We derive sublinear-time quantum algorithms for computing the Nash equilibrium of two-player zero-sum games, based on efficient Gibbs sampling methods. We are able to achieve speed-ups for both dense and sparse payoff matrices at the cost of a mildly increased dependence on the additive error compared to classical algorithms. In particular we can find $\varepsilon$-approximate Nash equilibrium strategies in complexity $\tilde{O}(\sqrt{n+m}/\varepsilon^3)$ and $\tilde{O}(\sqrt{s}/\varepsilon^{3.5})$ respectively, where $n\times m$ is the size of the matrix describing the game and $s$ is its sparsity. Our algorithms use the LP formulation of the problem and apply techniques developed in recent works on quantum SDP-solvers. We also show how to reduce general LP-solving to zero-sum games, resulting in quantum LP-solvers that have complexities $\tilde{O}(\sqrt{n+m}\gamma^3)$ and $\tilde{O}(\sqrt{s}\gamma^{3.5})$ for the dense and sparse access models respectively, where $\gamma$ is the relevant "scale-invariant" precision parameter